The Velocity of an Object Is the Distance It Travels Per Unit Time. Suppose the Velocity of a Bird

Learning Objectives

By the terminate of this section, yous will be able to:

  • Apply principles of vector addition to determine relative velocity.
  • Explain the significance of the observer in the measurement of velocity.

Relative Velocity

If a person rows a boat across a rapidly flowing river and tries to head directly for the other shore, the boat instead moves diagonally relative to the shore, as in Figure 1. The boat does not move in the direction in which it is pointed. The reason, of course, is that the river carries the boat downstream. Similarly, if a small airplane flies overhead in a potent crosswind, yous tin can sometimes meet that the plane is non moving in the direction in which it is pointed, as illustrated in Figure 2. The plane is moving straight ahead relative to the air, but the movement of the air mass relative to the footing carries it sideways.

A boat is trying to cross a river. Due to the velocity of river the path traveled by boat is diagonal. The velocity of boat v boat is in positive y direction. The velocity of river v river is in positive x direction. The resultant diagonal velocity v total which makes an angle of theta with the horizontal x axis is towards north east direction.

Effigy 1. A boat trying to head directly beyond a river volition actually move diagonally relative to the shore equally shown. Its total velocity (solid pointer) relative to the shore is the sum of its velocity relative to the river plus the velocity of the river relative to the shore.

An airplane is trying to fly straight north with velocity v sub p. Due to wind velocity v sub w in south west direction making an angle theta with the horizontal axis, the plane's total velocity is thirty eight point 0 meters per seconds oriented twenty degrees west of north.

Figure 2. An airplane heading straight north is instead carried to the west and slowed downwards by wind. The plane does not motility relative to the ground in the direction it points; rather, it moves in the management of its total velocity (solid pointer).

In each of these situations, an object has a velocity relative to a medium (such equally a river) and that medium has a velocity relative to an observer on solid ground. The velocity of the object relative to the observer is the sum of these velocity vectors, every bit indicated in Figure 1 and Figure two. These situations are only two of many in which information technology is useful to add velocities. In this module, we start re-examine how to add together velocities and so consider certain aspects of what relative velocity means. How practise nosotros add velocities? Velocity is a vector (it has both magnitude and management); the rules of vector addition discussed in Vector Add-on and Subtraction: Graphical Methods and Vector Addition and Subtraction: Belittling Methods apply to the addition of velocities, just as they do for any other vectors. In ane-dimensional move, the addition of velocities is simple—they add like ordinary numbers. For instance, if a field hockey player is moving at v m/s directly toward the goal and drives the ball in the aforementioned direction with a velocity of xxx grand/southward relative to her body, then the velocity of the brawl is 35 g/s relative to the stationary, profusely sweating goalkeeper standing in front of the goal. In ii-dimensional motion, either graphical or analytical techniques can be used to add velocities. We will concentrate on analytical techniques. The following equations give the relationships betwixt the magnitude and direction of velocity (v and θ) and its components (five x and v y ) along the x – and y -axes of an accordingly chosen coordinate arrangement:

v x = five cosθ

v y = five sinθ

[latex]v=\sqrt{{{v}_{x}}^{2}+{{v}_{y}}^{two}}[/latex]

θ = tan − one (v y /v x ).

The figure shows components of velocity v in horizontal x axis v x and in vertical y axis v y. The angle between the velocity vector v and the horizontal axis is theta.

Figure three. The velocity, v , of an object traveling at an angle θ to the horizontal axis is the sum of component vectors v x and v y.

These equations are valid for any vectors and are adapted specifically for velocity. The first ii equations are used to find the components of a velocity when its magnitude and direction are known. The last two are used to discover the magnitude and direction of velocity when its components are known.

Accept-Domicile Experiment: Relative Velocity of a Gunkhole

Make full a bathtub half-full of water. Have a toy boat or some other object that floats in water. Unplug the bleed so water starts to drain. Effort pushing the boat from one side of the tub to the other and perpendicular to the flow of water. Which way exercise you need to push the boat so that it ends up immediately opposite? Compare the directions of the menstruum of water, heading of the gunkhole, and actual velocity of the boat.

Example 1. Adding Velocities: A Boat on a River

A boat is trying to cross a river. Due to the velocity of the river the path traveled by the boat is diagonal. The velocity of the boat, v boat, is equal to zero point seven five meters per second and is in positive y direction. The velocity of the river, v-river, is equal to one point two meters per second and is in positive x direction. The resultant diagonal velocity v total, which makes an angle of theta with the horizontal x axis, is towards north east direction.

Figure four. A boat attempts to travel straight across a river at a speed 0.75 m/s. The current in the river, even so, flows at a speed of 1.20 m/s to the correct. What is the total displacement of the boat relative to the shore?

Refer to Figure four which shows a boat trying to go directly across the river. Let united states calculate the magnitude and direction of the boat's velocity relative to an observer on the shore, vtot. The velocity of the boat, fiveboat, is 0.75 grand/due south in the y -direction relative to the river and the velocity of the river, vriver, is i.20 m/southward to the correct.

Strategy

We start by choosing a coordinate system with its x-axis parallel to the velocity of the river, as shown in Figure iv. Because the gunkhole is directed straight toward the other shore, its velocity relative to the h2o is parallel to the y-axis and perpendicular to the velocity of the river. Thus, we can add together the two velocities by using the equations [latex]{v}_{\text{tot}}=\sqrt{{{v}_{x}}^{ii}+{{five}_{y}}^{2}}[/latex] and θ= tan−1(v y /v x ) directly.

Solution

The magnitude of the total velocity is

[latex]{v}_{\text{tot}}=\sqrt{{{v}_{x}}^{2}+{{five}_{y}}^{ii}}[/latex]

where

v x = v river = 1.xx thou/s

and

five y =five boat= 0.750 m/southward.

Thus,

[latex]{five}_{\text{tot}}=\sqrt{({1.xx}\text{ m/due south})^{ii} + ({0.750}\text{ one thousand/s})^{ii}}[/latex]

yielding

five tot= one.42 thousand/s.

The direction of the total velocity θ is given past:

θ= tan−1(5 y /v x ) = tan−1(0.750/1.20).

This equation gives

θ= 32.0º.

Discussion

Both the magnitude v and the direction θ of the total velocity are consequent with Figure 4. Note that because the velocity of the river is big compared with the velocity of the boat, it is swept rapidly downstream. This consequence is evidenced by the small angle (but 32.0º) the total velocity has relative to the riverbank.

Example 2. Calculating Velocity: Current of air Velocity Causes an Airplane to Migrate

Calculate the air current velocity for the situation shown in Effigy 5. The plane is known to exist moving at 45.0 m/due south due due north relative to the air mass, while its velocity relative to the ground (its total velocity) is 38.0 m/s in a direction 20.0º west of north.

An airplane is trying to fly north with velocity v p equal to forty five meters per second at angle of one hundred and ten degrees but due to wind velocity v w in south west direction making an angle theta with the horizontal axis it reaches a position in north west direction with resultant velocity v total equal to thirty eight meters per second and the direction is twenty degrees west of north.

Figure 5. An airplane is known to be heading northward at 45.0 m/south, though its velocity relative to the basis is 38.0 grand/s at an angle west of north. What is the speed and management of the wind?

Strategy

In this problem, somewhat unlike from the previous example, nosotros know the full velocity fivetot and that information technology is the sum of two other velocities, vw (the air current) and vp (the plane relative to the air mass). The quantity 5p is known, and nosotros are asked to detect fivew. None of the velocities are perpendicular, simply it is possible to find their components along a common set of perpendicular axes. If we can find the components of 5w, then we can combine them to solve for its magnitude and management. As shown in Figure 5, we choose a coordinate organization with its x-axis due east and its y-axis due north (parallel to fivep). (Yous may wish to look back at the discussion of the addition of vectors using perpendicular components in Vector Improver and Subtraction: Belittling Methods.)

Solution

Because vtot is the vector sum of the vwest and vp, its x– and y-components are the sums of the x– and y-components of the wind and plane velocities. Annotation that the aeroplane only has vertical component of velocity so v p10 = 0 and v py =five p. That is,

v tot x = 5 w x

and

five toty =v westwardx +v p.

We can apply the first of these ii equations to find v wx :

5 wx =5 totx =five totcos 110º.

Because v tot = 38.0 m/s and cos 110º=–0.342 nosotros have

5 wx = (38.0 thousand/s)(–0.342) = –13.0 k/s.

The minus sign indicates motility west which is consistent with the diagram. Now, to observe 5 wy we note that

v tot y = five west x + 5 p

Here v toty = five totsin 110º; thus,

v wy = (38.0 m/s)(0.940)−45.0 grand/southward = −9.29 1000/due south.

This minus sign indicates motion southward which is consistent with the diagram. Now that the perpendicular components of the wind velocity five wx and v westwardy are known, we can detect the magnitude and direction of fivew. Start, the magnitude is

[latex]\begin{assortment}{c}{{5}_{w}}\hfill=\hfill\sqrt{{{five}_{wx}}^{2}+{{five}_{wy}}^{2}} \\\hfill=\hfill\sqrt{({-13.0}\text{ grand/s})^{2} + ({-nine.29}\text{ m/s}^{2})}\end{assortment}[/latex]

so that

v due west= 16.0 1000/s.

The direction is:

θ = tan− 1 (v westy /v westward10 ) = tan− 1(−9.29/−13.0)

giving

θ= 35.6º.

Discussion

The wind's speed and direction are consequent with the significant consequence the current of air has on the total velocity of the plane, equally seen in Effigy v. Considering the aeroplane is fighting a strong combination of crosswind and head-wind, it ends upward with a total velocity significantly less than its velocity relative to the air mass besides as heading in a unlike direction.

Note that in both of the last 2 examples, nosotros were able to brand the mathematics easier by choosing a coordinate system with i centrality parallel to one of the velocities. We will repeatedly find that choosing an appropriate coordinate organisation makes problem solving easier. For example, in projectile motility we always use a coordinate organisation with one centrality parallel to gravity.

Relative Velocities and Classical Relativity

When adding velocities, we have been careful to specify that the velocity is relative to some reference frame . These velocities are called relative velocities. For example, the velocity of an aeroplane relative to an air mass is different from its velocity relative to the ground. Both are quite different from the velocity of an aeroplane relative to its passengers (which should be close to cypher). Relative velocities are ane aspect of relativity, which is defined to be the study of how different observers moving relative to each other measure the same phenomenon.

Nearly everyone has heard of relativity and immediately assembly it with Albert Einstein (1879–1955), the greatest physicist of the 20th century. Einstein revolutionized our view of nature with his mod theory of relativity, which we shall report in later chapters. The relative velocities in this section are actually aspects of classical relativity, commencement discussed correctly by Galileo and Isaac Newton. Classical relativity is limited to situations where speeds are less than well-nigh 1% of the speed of light—that is, less than 3,000 km/s. Most things we encounter in daily life movement slower than this speed.

Allow us consider an case of what two different observers see in a situation analyzed long ago past Galileo. Suppose a sailor at the top of a mast on a moving transport drops his binoculars. Where volition it hit the deck? Will it striking at the base of the mast, or will it hit behind the mast because the transport is moving forward? The answer is that if air resistance is negligible, the binoculars volition striking at the base of the mast at a betoken directly beneath its point of release. Now let united states consider what two different observers encounter when the binoculars drib. One observer is on the ship and the other on shore. The binoculars accept no horizontal velocity relative to the observer on the ship, and then he sees them autumn directly downwards the mast. (Run into Effigy half-dozen.) To the observer on shore, the binoculars and the ship have the same horizontal velocity, so both move the same altitude forward while the binoculars are falling. This observer sees the curved path shown in Figure 6. Although the paths await dissimilar to the different observers, each sees the same result—the binoculars hit at the base of operations of the mast and not behind it. To become the right description, information technology is crucial to correctly specify the velocities relative to the observer.

A person is observing a moving ship from the shore. Another person is on top of ship's mast. The person in the ship drops binoculars and sees it dropping straight. The person on the shore sees the binoculars taking a curved trajectory.

Effigy half dozen. Classical relativity. The same motility equally viewed past two different observers. An observer on the moving ship sees the binoculars dropped from the top of its mast fall straight down. An observer on shore sees the binoculars take the curved path, moving forward with the ship. Both observers run across the binoculars strike the deck at the base of the mast. The initial horizontal velocity is dissimilar relative to the ii observers. (The ship is shown moving rather fast to emphasize the consequence.)

Case iii. Computing Relative Velocity: An Airline Rider Drops a Money

An airline passenger drops a coin while the aeroplane is moving at 260 m/s. What is the velocity of the money when it strikes the flooring i.l chiliad below its point of release: (a) Measured relative to the plane? (b) Measured relative to the Earth?

A person standing on ground is observing an airplane. Inside the airplane a woman is sitting on seat. The airplane is moving in the right direction. The woman drops the coin which is vertically downwards for her but the person on ground sees the coin moving horizontally towards right.

Effigy 7. The motion of a coin dropped inside an airplane as viewed by two dissimilar observers. (a) An observer in the airplane sees the coin fall straight down. (b) An observer on the footing sees the money motility nigh horizontally.

Strategy

Both problems can be solved with the techniques for falling objects and projectiles. In role (a), the initial velocity of the coin is zero relative to the plane, so the motion is that of a falling object (one-dimensional). In part (b), the initial velocity is 260 m/s horizontal relative to the Globe and gravity is vertical, and then this motion is a projectile movement. In both parts, it is best to apply a coordinate system with vertical and horizontal axes.

Solution for (a)

Using the given information, we note that the initial velocity and position are nada, and the final position is ane.50 k. The terminal velocity can be constitute using the equation:

v y two = v 0y ii −21000(yy 0).

Substituting known values into the equation, nosotros go

[latex]{{5}_{y}}^{ii}={0}^{ii}-2\left(9\text{.}\text{80}{\text{g/south}}^{two}\right)\left(-one\text{.}\text{l}\text{chiliad}-0 m\correct)=\text{29}\text{.}four{\text{g}}^{ii}{\text{/south}}^{2}[/latex]

yielding

v y = −v.42 grand/s.

We know that the square root of 29.4 has two roots: 5.42 and -five.42. We choose the negative root because we know that the velocity is directed downwards, and we have defined the positive direction to be upwardly. At that place is no initial horizontal velocity relative to the plane and no horizontal dispatch, and so the motion is straight down relative to the aeroplane.

Solution for (b)

Considering the initial vertical velocity is zippo relative to the ground and vertical motion is independent of horizontal motion, the final vertical velocity for the coin relative to the ground is vy = -5.42 k/s, the same as found in part (a). In contrast to part (a), there now is a horizontal component of the velocity. However, since there is no horizontal acceleration, the initial and final horizontal velocities are the same and five x =260 m/south. The x– and y-components of velocity can be combined to observe the magnitude of the final velocity:

[latex]five=\sqrt{{{v}_{x}}^{2}+{{v}_{y}}^{ii}}[/latex].

Thus,

[latex]five=\sqrt{({260\text{ m/s})}^{2}+({-5.42\text{ 1000/s}})^{two}}[/latex]

yielding

v= 260.06 g/s.

The direction is given by:

θ = tan−1(v y /v x ) = tan−1(−5.42/260)

so that

θ= tan−1(−0.0208)=−1.19º.

Discussion

In part (a), the terminal velocity relative to the airplane is the same as it would be if the coin were dropped from rest on the Earth and barbarous 1.l m. This result fits our experience; objects in a aeroplane fall the aforementioned way when the airplane is flying horizontally every bit when it is at rest on the ground. This effect is also true in moving cars. In function (b), an observer on the basis sees a much different motion for the coin. The plane is moving so fast horizontally to begin with that its final velocity is barely greater than the initial velocity. Once over again, we see that in two dimensions, vectors practise non add like ordinary numbers—the terminal velocity 5 in office (b) is not (260 – 5.42) m/due south; rather, it is 260.06 m/s. The velocity's magnitude had to be calculated to v digits to see any difference from that of the airplane. The motions equally seen by different observers (one in the plane and 1 on the footing) in this example are coordinating to those discussed for the binoculars dropped from the mast of a moving transport, except that the velocity of the plane is much larger, then that the 2 observers run into very dissimilar paths. (See Figure 7.) In addition, both observers run into the coin autumn 1.l m vertically, simply the one on the ground too sees information technology motion forward 144 thou (this adding is left for the reader). Thus, 1 observer sees a vertical path, the other a nigh horizontal path.

Making Connections: Relativity and Einstein

Because Einstein was able to conspicuously define how measurements are made (some involve light) and because the speed of low-cal is the aforementioned for all observers, the outcomes are spectacularly unexpected. Fourth dimension varies with observer, energy is stored as increased mass, and more surprises wait.

PhET Explorations: Motion in 2D

Endeavor the new "Ladybug Motion 2nd" simulation for the latest updated version. Learn about position, velocity, and acceleration vectors. Move the ball with the mouse or let the simulation movement the brawl in four types of motion (two types of linear, simple harmonic, circle).

Motion in 2D

Click to download. Run using Java.

Summary

  • Velocities in two dimensions are added using the same analytical vector techniques, which are rewritten as

    v x = v cosθ

    five y = 5 sinθ

    [latex]five=\sqrt{{{v}_{x}}^{2}+{{v}_{y}}^{2}}[/latex]

    θ = tan − 1 (v y /v x ).

  • Relative velocity is the velocity of an object as observed from a particular reference frame, and information technology varies dramatically with reference frame.
  • Relativity is the study of how different observers measure the aforementioned phenomenon, peculiarly when the observers move relative to one another. Classical relativity is limited to situations where speed is less than virtually ane% of the speed of low-cal (3000 km/due south).

Conceptual Questions

  1. What frame or frames of reference practice you lot instinctively use when driving a car? When flying in a commercial jet plane?
  2. A basketball player dribbling downward the court commonly keeps his eyes fixed on the players around him. He is moving fast. Why doesn't he demand to keep his eyes on the ball?
  3. If someone is riding in the dorsum of a pickup truck and throws a softball direct astern, is it possible for the ball to autumn straight down as viewed by a person standing at the side of the route? Nether what condition would this occur? How would the movement of the ball appear to the person who threw it?
  4. The hat of a jogger running at constant velocity falls off the back of his caput. Draw a sketch showing the path of the hat in the jogger's frame of reference. Draw its path as viewed by a stationary observer.
  5. A clod of dirt falls from the bed of a moving truck. It strikes the ground straight below the finish of the truck. What is the direction of its velocity relative to the truck just before it hits? Is this the aforementioned as the direction of its velocity relative to ground only before it hits? Explain your answers.

Issues & Exercises

1. Bryan Allen pedaled a human-powered aircraft across the English language Channel from the cliffs of Dover to Cap Gris-Nez on June 12, 1979. (a) He flew for 169 min at an average velocity of 3.53 yard/due south in a management 45º due south of eastward. What was his total deportation? (b) Allen encountered a headwind averaging 2.00 k/s most precisely in the opposite direction of his movement relative to the Globe. What was his average velocity relative to the air? (c) What was his full displacement relative to the air mass?

2. A seagull flies at a velocity of ix.00 one thousand/southward straight into the current of air. (a) If it takes the bird 20.0 min to travel 6.00 km relative to the Earth, what is the velocity of the wind? (b) If the bird turns around and flies with the current of air, how long will he take to return 6.00 km? (c) Talk over how the wind affects the total circular-trip fourth dimension compared to what it would exist with no current of air.

iii. Near the end of a marathon race, the first two runners are separated by a distance of 45.0 m. The front end runner has a velocity of 3.50 m/s, and the second a velocity of four.20 m/s. (a) What is the velocity of the second runner relative to the first? (b) If the front runner is 250 g from the finish line, who will win the race, bold they run at abiding velocity? (c) What distance ahead will the winner be when she crosses the finish line?

4. Verify that the coin dropped by the airline passenger in Figure 7 travels 144 chiliad horizontally while falling i.50 k in the frame of reference of the Earth.

5. A football game quarterback is moving straight backward at a speed of 2.00 m/s when he throws a pass to a player eighteen.0 m straight downfield. The brawl is thrown at an bending of 25.0º relative to the ground and is caught at the same tiptop as it is released. What is the initial velocity of the ball relative to the quarterback ?

6. A ship sets canvass from Rotterdam, The Netherlands, heading due northward at 7.00 g/s relative to the water. The local bounding main electric current is one.fifty thousand/s in a direction 40.0º north of east. What is the velocity of the ship relative to the Earth?

7. (a) A jet aeroplane flight from Darwin, Australia, has an air speed of 260 m/s in a direction 5.0º south of west. Information technology is in the jet stream, which is bravado at 35.0 m/southward in a direction 15º s of e. What is the velocity of the airplane relative to the Earth? (b) Discuss whether your answers are consistent with your expectations for the effect of the wind on the airplane's path.

eight. (a) In what management would the ship in Practise half dozen have to travel in order to have a velocity straight north relative to the Earth, assuming its speed relative to the water remains 7.00 m/s? (b) What would its speed exist relative to the World?

9. (a) Another aeroplane is flying in a jet stream that is blowing at 45.0 m/s in a direction 20º south of e (as in Exercise vii). Its direction of motility relative to the World is 45º south of west, while its direction of travel relative to the air is five.00º s of west. What is the aeroplane's speed relative to the air mass? (b) What is the plane's speed relative to the World?

10. A sandal is dropped from the pinnacle of a 15.0-m-high mast on a send moving at 1.75 m/s due south. Calculate the velocity of the sandal when it hits the deck of the ship: (a) relative to the ship and (b) relative to a stationary observer on shore. (c) Discuss how the answers give a consistent result for the position at which the sandal hits the deck.

11. The velocity of the air current relative to the water is crucial to sailboats. Suppose a sailboat is in an ocean current that has a velocity of 2.xx m/s in a direction 30.0º east of north relative to the Earth. It encounters a wind that has a velocity of 4.fifty k/s in a direction of 50.0º southward of due west relative to the Globe. What is the velocity of the wind relative to the h2o?

12. The nifty astronomer Edwin Hubble discovered that all afar galaxies are receding from our Milky Way Milky way with velocities proportional to their distances. It appears to an observer on the Earth that we are at the heart of an expanding universe. Effigy ix illustrates this for five galaxies lying along a direct line, with the Milky way Galaxy at the center. Using the data from the effigy, calculate the velocities: (a) relative to galaxy 2 and (b) relative to galaxy 5. The results mean that observers on all galaxies volition see themselves at the center of the expanding universe, and they would likely be aware of relative velocities, concluding that information technology is not possible to locate the middle of expansion with the given information.

Five galaxies on a horizontal straight line are shown. The left most galaxy one has distance of three hundred millions of light years and it is moving towards left. The second and third galaxies in the figure have shown no velocities. The velocities of fourth and fifth galaxies are towards right.

Effigy 9. Five galaxies on a straight line, showing their distances and velocities relative to the Galaxy (MW) Galaxy. The distances are in millions of lite years (Mly), where a light year is the distance calorie-free travels in one twelvemonth. The velocities are most proportional to the distances. The sizes of the galaxies are greatly exaggerated; an average milky way is about 0.ane Mly across.

(a) Use the distance and velocity data in [link] to find the rate of expansion as a function of altitude.

(b) If you lot extrapolate back in time, how long ago would all of the galaxies accept been at approximately the same position? The two parts of this problem give y'all some idea of how the Hubble constant for universal expansion and the time back to the Big Bang are adamant, respectively.

thirteen. An athlete crosses a 25-thousand-wide river past swimming perpendicular to the water current at a speed of 0.5 m/due south relative to the water. He reaches the opposite side at a distance 40 thou downstream from his starting indicate. How fast is the h2o in the river flowing with respect to the ground? What is the speed of the swimmer with respect to a friend at balance on the ground?

xiv. A ship sailing in the Gulf Stream is heading 25.0º westward of north at a speed of 4.00 m/south relative to the h2o. Its velocity relative to the Earth is 4.80 one thousand/s 5.00º due west of north. What is the velocity of the Gulf Stream? (The velocity obtained is typical for the Gulf Stream a few hundred kilometers off the east coast of the Usa.)

15. An water ice hockey player is moving at 8.00 one thousand/s when he hits the puck toward the goal. The speed of the puck relative to the player is 29.0 m/s. The line between the center of the goal and the player makes a 90.0º angle relative to his path as shown in Figure ten. What angle must the puck's velocity make relative to the player (in his frame of reference) to hitting the center of the goal?

An ice hockey player is moving across the rink with velocity v player towards north direction. The goal post is in east direction. To hit the goal the hockey player must hit with velocity of puck v puck making an angle theta with the horizontal axis so that its direction is towards south east.

Figure 10. An ice hockey player moving across the rink must shoot backward to requite the puck a velocity toward the goal.

16. Unreasonable Results Suppose you wish to shoot supplies straight upwardly to astronauts in an orbit 36,000 km above the surface of the Earth. (a) At what velocity must the supplies be launched? (b) What is unreasonable well-nigh this velocity? (c) Is at that place a trouble with the relative velocity betwixt the supplies and the astronauts when the supplies reach their maximum height? (d) Is the premise unreasonable or is the available equation extraneous? Explain your respond.

17. Unreasonable Results A commercial plane has an air speed of 280 m/due south east and flies with a stiff tailwind. It travels 3000 km in a management 5º south of due east in 1.50 h. (a) What was the velocity of the plane relative to the basis? (b) Summate the magnitude and direction of the tailwind'southward velocity. (c) What is unreasonable near both of these velocities? (d) Which premise is unreasonable?

18. Construct Your Own Problem Consider an plane headed for a runway in a cross current of air. Construct a problem in which you calculate the angle the plane must fly relative to the air mass in order to have a velocity parallel to the rails. Among the things to consider are the direction of the track, the wind speed and direction (its velocity) and the speed of the plane relative to the air mass. Also calculate the speed of the airplane relative to the footing. Discuss any last minute maneuvers the airplane pilot might have to perform in order for the aeroplane to land with its wheels pointing straight down the runway.

Glossary

classical relativity:
the study of relative velocities in situations where speeds are less than about one% of the speed of calorie-free—that is, less than 3000 km/s
relative velocity:
the velocity of an object equally observed from a particular reference frame
relativity:
the study of how different observers moving relative to each other measure the same phenomenon
velocity:
speed in a given management
vector improver:
the rules that apply to calculation vectors together

Selected Solutions to Problems & Exercises

1. (a) 35.eight km, 45º south of east (b) 5.53 m/s, 45º south of east (c) 56.1 km, 45º south of due east

3. (a) 0.70 1000/s faster (b) 2d runner wins (c) four.17 k

5. 17.0 1000/s, 22.1º

7.  (a) 230 yard/s, 8.0º southward of w (b) The wind should make the plane travel slower and more to the s, which is what was calculated.

9. (a) 63.5 m/s (b) 29.half-dozen m/s

xi. 6.68 1000/s, 53.3º southward of west

12.  (a) [latex]{H}_{\text{average}}=\text{14}\text{.}\text{9}\frac{\text{km/s}}{\text{Mly}}[/latex] (b) 20.2 billion years

14. 1.72 m/s, 42.3º north of east

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